I have the following exact sequence $\mathbb{Z}\xrightarrow{f}\mathbb{Z} \xrightarrow{g} K_0(\mathcal{T})\xrightarrow{h}\mathbb{Z}\xrightarrow{0}0$.
From here I want to conclude that $K_0(\mathcal{T})\cong \mathbb{Z}$.
So far what I have is that since h is surjective, $K_0(\mathcal{T})$ is at least as big as $\mathbb{Z}$. But how do I conclude that g or h is a bijection? I just never worked with exact sequences and really stuck here.
Thank you, everyone.
Additional information: This is the attempt to compute the K-groups of Toeplitz algebra from the exact sequence given. This is a problem in Wegge-Olsen book in Ch6. The complete sequence is $0\to K_1(\mathcal{T})\to\mathbb{Z}\xrightarrow{f}\mathbb{Z} \xrightarrow{g} K_0(\mathcal{T})\xrightarrow{h}\mathbb{Z}\xrightarrow{0}0$. From here, I easily concluded that $K_1(\mathcal{T})=0$.
The information given is insufficient to conclude anything more than what you have already concluded without invoking conditionals. The problem is that $f$ can be anything; no restrictions are imposed by your setup. So you can't exclude the possibility that $K_0$ is of rank 2 or possesses nontrivial torsion. You need more information.